MathDB

23

Part of 2011 Sharygin Geometry Olympiad

Problems(1)

line cuts a triangle, midpoints of segments of projections of intersections

Source: 2011 Sharygin Geometry Olympiad Correspondence Round P23

4/7/2019
Given are triangle ABCABC and line ℓ\ell intersecting BC,CABC, CA and ABAB at points A1,B1A_1, B_1 and C1C_1 respectively. Point A′A' is the midpoint of the segment between the projections of A1A_1 to ABAB and ACAC. Points B′B' and C′C' are defined similarly. (a) Prove that A′,B′A', B' and C′C' lie on some line ℓ′\ell'. (b) Suppose ℓ\ell passes through the circumcenter of △ABC\triangle ABC. Prove that in this case ℓ′\ell' passes through the center of its nine-points circle.
M. Marinov and N. Beluhov
geometryNine Point CircleprojectionsmidpointsCircumcentercollinear